L13-elastic_settlement_calculation

# L13-elastic_settlement_calculation - 1 13 SETTLEMENT OF...

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13. SETTLEMENT OF STRUCTURES 13.1 Solutions based on the theory of elasticity Figure 1 represents a surface footing resting on a soil layer of depth H. Soil Layer Rigid bedrock H P The settlement, s, of any point can be determined from s dz zz H = ∆ε 0 (1a) where for an elastic soil ∆ε zz zz xx yy zz E = + - + + ( ) ( ) 1 ν σ ν σ σ σ (1b) and under undrained conditions: ∆ε ∆σ ∆σ ∆σ ∆σ zz u zz u xx yy zz u E = + - + + ( ) ( ) 1 ν ν (1c) As discussed earlier, to determine the settlement immediately after the application of the load equation (1c) is used, and to determine the long term or drained settlement equation (1b) is used. In the latter case the changes in pore water pressure u are usually zero and so the increment in effective stress is equal to the increment in total stress. Thus, in both cases the settlement can be calculated if both the change in total vertical stress ∆σ zz and the change in the mean total stress ( ∆σ xx + ∆σ yy + ∆σ zz ) are known. It has been shown previously how the Boussinesq solution for the stresses in an elastic half space due to a point load acting on the surface can be used to determine the stress distributions under a 1 Fig. 1 Foundation resting on a soil layer

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variety of shapes of loaded areas (circles, rectangles, arbitrary shapes). The same solution can be used to determine the surface settlements, s r as a function of the distance, r, from a point load Q, as s Q Er r = - ( ) 1 2 ν π (2) This is illustrated in Figure 2. Q r s r Fig. 2 Surface deflection of a deep elastic layer s Q Er r = - ( ) 1 2 ν π Because the soil is assumed to be linear elastic it is possible to use superposition to determine the surface settlements for distributed loads using the point load solution. For example, the settlement at the centre of a circular loaded area, radius, a, with uniform stress, q, (flexible foundation), can be determined by considering the effect of the stress, q, acting over an area r d θ dr (shown in Figure 3) on the settlement at the centre. The settlement is then given by: dr r d θ d θ Fig. 3 Stress q acting over a circular area of radius a s Er qrd dr q a E centre a = - = - ( ) ( ) 1 2 1 2 0 2 0 2 ν π θ ν π (3) 2 Fig. 2 Surface deflection due to a point load on a deep elastic layer H a
For other positions under the circular load and for other shapes the integration is not so straightforward, and in many cases analytical solutions will not be possible. Also a limitation of this (Boussinesq) solution is that it assumes the soil layer is infinitely deep. This rarely occurs in practice as more generally a relatively shallow soil layer usually overlies rock. The procedure adopted in practice is to make use of charted solutions that are available for a number of commonly encountered situations. Some of these are given in the data sheets, and are discussed below. For other solutions the book "Elastic solutions for Soil and Rock Mechanics" by Poulos and Davis should be referred to.

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L13-elastic_settlement_calculation - 1 13 SETTLEMENT OF...

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